Bernstein Polynomials

Bernstein polynomials are a remarkable family of polynomials associated to any given function on the unit interval. Their first notable appearance was in Bernstein's proof of the Weierstrass approximation theorem. This book gives an exhaustive exposition of the main facts about the Bernstein polynomials and discusses some of their applications in analysis. The first three chapters of the book give an introduction to a theory of singular integrals by means of the particular instance of Bernstein polynomials. The author writes in the preface to this second edition, ""After the trigonometric integrals, Bernstein polynomials are the most important and interesting concrete operators on a space of continuous functions. Since the appearance of the first edition of this book [in 1953], the interest in this subject has continued. In an appendix we have summed up a few of the most important papers that have appeared since.''

Table of Contents:
Introduction Bernstein Polynomials in Real Domain: 1.1 The theorem of Weierstrass; 1.2 Other proofs of the theorem of Weierstrass; 1.3 Generalizations of the theorem of Weierstrass; 1.4 Derivatives of the Bernstein polynomials; 1.5 Estimations and lemmas; 1.6 The degree of approximation by Bernstein polynomials; 1.6.1 Asymptotic formulae; 1.7 Monotone functions, convex functions, and functions of bounded variation; 1.8 Further theorems on derivatives; 1.9 Discontinuous functions Generalizations of Bernstein Polynomials: 2.1 Approximation of integrable functions; 2.2 Approximation of measurable functions; 2.3 Bernstein polynomials on an unbounded interval; 2.4 General methods of summation; 2.5 Euler methods of summation; 2.6 Degenerate Bernstein polynomials; 2.7 Divided differences and generalized $p_{n\nu}(x)$; 2.8 Approximation by linear aggregates of functions $x^{\alpha}$; 2.9 Some further generalizations Spaces of Functions and Moment Problems: 3.1 Banach spaces; 3.2 Functionals and moment problems; 3.3 Moment problems with Stieltjes integrals; 3.4 Rearrangements of functions; 3.5 Spaces $\Lambda$ and $M$. Other spaces of integrable functions; 3.6 Inequalities for spaces $\Lambda$; 3.7 Continuous linear functionals in spaces $\Lambda$ and $\Lambda^*$; 3.7.1 Spaces $M(\phi, p)$ as conjugate spaces; 3.8 Moment problems for integrable functions; 3.9 Hausdorff methods of summation Bernstein Polynomials of Analytic Functions: 4.1 Preliminary theorems; 4.2 Contour integrals and asymptotic formulae; 4.3 The loop $L_z$ and its properties; 4.3.1 Further properties of the loop; 4.4 The fundamental convergence theorem; 4.5 Sets of convergence and autonomous sets; 4.6 Examples of autonomous sets and sets of convergence; 4.7 Functions analytic on a part of the interval $[0, 1]$; 4.8 Summation of power series; 4.9 Degenerate Bernstein polynomials in the complex domain Bibliography Index